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CHAPTER II . I N TERACTION OF RADIATION WITH MATTER 
k. ENERGY LOSS BY CHARGED PARTICLES 



1, A charged particle moving through matter loses energy 
by electromagnetic interactions which raise electrons of the 
matter to excited energy states. If an excited level is in the 
continuum of states the electron is Ionised; if not, the electron 
Is in an excited bound state. In either case the Increment of 
energy is taken from the kinetic energy of the incident particle. 
In the following section "ionization" will refer to both degrees 
of excitation. 

Range - total distance traveled by the particle until its 
kinetic energy is 0. Before a formula for the range of a par- 
ticle can be derived, the rate of energy loss per unit path must 
be calculated. The first such calculation. Is due to Bohr*, and 
is essentially classical, I.e., non-quantum mechanical. 

2. Bohr Formula. Consider one electron of mass m at a 
distance b from the path of an incident particle having charge 
Je, mass M and velocity V. 

a (electron) 

— fo= Impact parameter 



mass M 
charge £e»- 
velocity V 



A 



FIG-. II. 1 



Assume the electron ts free and initially at rest, and 
moves so slightly during the collision that the electric field 
acting on the electron due to the particle can be calculated at 
the initial location of the electron. The last assumption Is 
not valid for an Incident particle of velocity comparable to 
that acquired by the electron. 

We shall calculate first the momentum acquired by an elec- 
tron during a collision, and from this find the energy acquired. 
As the particle passes, the electrostatic force F changes 
direction. By symmetry the Impulse Jf (1 dLt parallel to the 

path is zero, since for each position of the particle to the 
left of A, yielding a forward contribution to the impulse, there 
Is a position at equal distance to the right of A giving an equal 
but opposite contribution. 



t-J 



F, dt 



We first estimate 



,*e' b 



The impulse -L to the path is 
the order of magnitude of %± '■ 

If. = J Fi at ^(electrostatic force )X(time of collision)--^- Ky" 

More exact computation: Consider a circular" cylinder 
centered on the path and passing through the position of the 
electron, Fig. II. 2, Let S be the electrostatic field in- 
tensity due to the particle VS. The electric flux Is 



Js-ds = 4Tr^e 



(independent of V) 



* Bohr, Phil, Mag . 24, 10 (1913)', 10, 581 (1915) 



2? 



2C 



Bohr Formula 



Ch. II 



toy Gauss's theorem. If S x = component of £ _L to the path, 
then the flux = f% Zv ktl = "tff^ft. Therefore f^jisf^^S , 

The variation of £1 with time at the electron Is the same 
function as would toe found by keeping J-e fixed and observing at 
a point moving with velocity V along the cylinder surface. 
Therefore 



T\ib 



T 



FIG. II. 2 

The Ins pulse Ii."i«, £j.€«-*~ yV = *'■■ 
momentum acquired by the electron, 
electron is then 

2_nrL omV 2 b z 



where p is ike II . ? 

The energy acquired by one 



II 



The number of collisions per unit path length such that b 
lies in the range to to b + db is equal to the number of electrons 
per cm, length in the shell bounded by cylinders of radii to and 
b + db (Fig. I 1.2). If % = number of electrons per cm3, this Is 
2n "ft bdb. The energy lost per cm. to such electrons is 



cLE(b) - — 



^4-7rfe + ^ ^b 



rm V' 



II. 4 



verge 



;m 



) 



II. 5 



The total change in energy to all shells in the range 
b min t0 b max lB 

The limit b max : The force on an electron as a function of 
time will be a pulse occupying a %±mp *%■=■ b/V , non-relativistically , 
It can be shown that if l/£ is much less than the vibration fre- 
quency 2/ of an electron In an atom, then the electron absorbs no 
energy, i.e., the probability for transition to a higher state Is 
small , 

Relativistic ally, the electric field of the incident particle 
Is contracted in the direction of motion and S± is increased by 



a factor 



Vf\ 



-/3'- 



A~% 



This "sharpens" the impulse given 



the electron; the duration 'V of the pulse of force Is now 
approximatelyY^^ifTrgF . The Integral of the Impulse is not 

changed, since it depends essentially on the product (field 
strength)X(duration) . 



Ch. II 
b T 



Bohr Formula 

1/ ^— _. b\|7^3= 



29 



Tjg^ is chosen so that /^ >7^ 
range integrated. Thus we may set 



b_ = 



V 



V 



(cm) 



is valid over the 



II. 6 



where W is an appropriate average freauency for electrons In 
the absorbing material. 



Problem: Discuss the statement that for \/ v <T, energy 
transfer to electrons Is negligible (or indole of adiabatic 
invariance) . 

Consider the component of the motion of the electron J- to 
the path of the incident particle. Let the coordinate of the 
electron be y. y = b + d sin^t. y = jy d cos V t. b»d. 

A (energy) = {j component of force )X( velocity ) dt 






Since cos 6^1 



r Cfb) 



© 



A( energy) < f^P^j^J^pq^dt = 



Vi 



T 

I 

1 



Y 2 -i,M &*+/<'' 






V 



a 






as 7^r->»i 



The limit b min : (l) Classically, the maximum velocity that 
can he imparted to the electron (in head-on collision) is less **~ 
than 2V.* The energy given cannot exceed 4m(2V) 2 . Therefore 
b cannot have values that imply a greater energy transfer per 
collision than 2mV- As a function of b, the energy transferred 
collision la |^ . Values of b smaller than the solution 



coll 

oer 



This determines b mln as 



must be excluded in the integration, 



ffotfl 



J£ 



[1.7 



' 



M i. (2 l + ^ hl f claGSlcal - treatment is valid only if the Coulomb 
field oi the incident particle varies negligibly over the atraeii- 
3i on ^ ot the quantum mechanical wave packet representing the 



* H 11 ? ls ea3 ie 8 "t to see in the rest system of the Incident par- 
ticle. Then the electron apoears to collide with something 
like a rigid wall. 






30 



Bohr Formula 



Gi- 



ll 



electron, k Is approximately the do Broglic wavelength of the 
electron as seen from the Incident particle. In a coordinate 
system In which the incident particle is at rest {this nearly 
coincides with the center of mass system , for a heavy Incident 
particle), the electron has velocity of about V, aBSUffilng its 
orbital velocity is much less than V. The momentum of the 
electron in this coordinate system is mtfj/sfiZg* and therefore 

&** ~^W • Only values of V>4\ have meaning, and therefore 

another criterion for b 



mm 



is 



k 



M3I 

ritiV 



II. 8 



The larger of (b Eln ) d and (b mln ) aM should be used in the 



integration*. For values of V 
and therefore II. 8 whould be used 



ere o max -> 
Using II ,.; 



•in* ( b min)> ( b minL 
i in II. 5: *" 






II. 9 



where V is a suitable average of the oscillation frequencies 
of the electrons, 

.More precise calculation** leads to the following formula 
for heavy particles, I.e., not electrons: 

r 






(erg cm ) 
(heavv particles) 



11.10 



where I is the average ionization potential of the electrons of 
the absorber, in ergs. The In term as 9 for 1 mev protons in 
NTP air . 

3. Electrons . There are two main reasons why 11.10 canr.ot 
apply to electrons. (l) The derivation assumes that the Incident 
particle Is practically undeflected. But the Incident particle 
acquires a transverse component of momentum per collision approx- 
imately equal to that given to an electron In the absorber, and 
if the incident particle Is an electron, the transverse velocity 
corresponding to" this momentum will not be negligible. (2) For 
collisions between Identical oarticles exchange phenomena must 
be taken into account*--*. Eethe *'# gives the following formula 
for energy loss by electrons: 



oLx/ lw . V^ t 2I 2 073 a ) s / } 

___^ . (electrons) 



11,11 



where I is the average Ionization potential of the atoms of the 
absorber and T = relativistic kinetic energy of the electron. 

*Ir. cutting the Integral of II. 4 off at \ ln ^O, we have neglected 
a term / bm ' T1 - cL E(b) . This is lust If led In "Lecture Series in Nuclear 
Physics", LA 24, Lecture XI, printed edition p. 2?. 

** Bethe, Handbuch der Fhysik , p. 519 

***Mott, Proc. Rov . Soc. 125 , 222, JK26 , 259 (1929) 



Ch. II 



Bohr Formula, Electron: 



Ari approximation for I ig-J? 



(■ergs) 



11.12 



val U ef; iv f ?fS%g-g b 2! Sl ^*™> ****** *> air, ^ the 



** -Ot her particles. For 
incident particles of identical 
charge moving in like absorbers 
-a_E/dx is a function of V bhlv 
Therefore if -dE/dx as a func- ' 
t-ion of enernv is Vatintfi for, say 
protons of mass y it can be 
tound for some other singly 
charged particle B by changing 
the energy scale so that the new 



Energy, in 
e . y . 

leu 

10° 

10 I 
108 

10 ^ 
10 10 



Electrons in Ai r 

., dE in e.v = per 

^X__^gZcm^ 

6 



19.5 x 
3.67 
1.69 
1.95 
2.47 
2.79 
3.48 



LQ 



energy values are K B /M D times 
the old. The following table 

Cl) de r uS?o 3 r.f TXV n 4w enatoleS ^ lllnS ° Ut thG table f °^ 
energy correction is ™a p v,,+ J i+\, lerent mass, the above 

valuers multiplied brrMf-^si "^ IT™^ l0SS 

* Uk/*/- > B faince fy enters the formula. 

ergs per ,ram cm^^ S^oted^ Gr l0GB ln thiS table is 

where f is the thickness in a- CB r a . Fo- thi* 
1S n °i : 'o ain3d ? r ?S ^ E ' 7df b ^ '"ultiolvin 



e rg~ c n: 
of air, 



11.13 

tab! e -dE/dx i n 
g by the density 



PROTONS" 



In Air 



Energy , _dE Mgy 




d^ 



-cm" 



300 
47 
7.6 
2.3 



^^^PP^X^^OlSi^^ 



e. v. 



2xl0 6 



JL 



cv 






jT e.v. 



.... 300 
2x101 47 

2xl0« 7.6 
2.3 
2.3 



2x10- 
2xlD 10 



- ,- -■ -- - — £U£LJ^L 

,117X10° 300 ,4x10^ 1200T106 iftQ 
.117x107 47 4xloI ' l "~"'- '■"' 

.117x108 7.6 4xl0 8 30 
.117x10- 2.3 UxlO 9 c 2 
.UTxlO 10 g,3|4 xl0 10 I 2 



188 lllOl 27,^ 

10 a 5 

10 9 ] .6 
ilO 



10^1.6 



portionaf^c^e^dfn-sTfv ot^fS^^^ l0SS W " , Xact ^ P™~ 
£" c ^ 2 ^r a given "articK ^ d SJ**** ^2 1^ e ^ POT 
on two further fee tors- m K Val 2' But " dE /^ denends 

? of an atom does no^ increaS T S^' e ^trons rer atom; 
in the formula for (ffi/dS J?l e 1, ?i ? ' ? 'f ths v * 1 £S ht -- thusj?^ 
appearing in the in - t=t > f heavier elements. (2} I 

C the -°'- term ' depends on the absorber (Eqn. 11.12)4*. 

Wx&rJt^ tSfs e ^« n . e f£ erl f eilts the original, data is the 
... na iron, tni. the energy is estimated. We have derived an 



See also Livingston and Sethe , Bev . Kod . Phys . 9, p. 26 5 . 



m 



Bohr Formula, Range 



Ch. II 



equation of tne for, -dE/dx = f{l). Integrating, we S et 



II .14 



X as the range of a particle wxwi * rough approximation 

nation ^y teP^formM numerically Fpr a ^ ug_ no ^ elatlvl ti _ 
we assume f(E ) °Cl/V^l/E. Ihen k • o ' bet ter approxiraa- 

cally. More precise consideration showB taai a 
tion Is* 



% _ w3 



U* Ef^V 



11.15 



jk___,o ».*.< A rourli formula Giving 
Empirical range- energy formulas . A roij ^ atmog pheric 



range of alpha particles in air at 15 



the 

oressure is 



36 



■ft*s.32 (Mev) £ C>vl. ( alph aB in air) 



II. 16 



IMa is correct to ahout 10% ™^$J^J^™%™£*te 

velocities. The general nature of tne range enei^y 

shown in Fig. IT»3- 




7 Poi-rization_^ffects^ In the T< 

derivaUon^f-IlTlO no ^^^.^f 
taken of the Influence on one electron 
due to the simultaneous motion of tne 
otner electrons near it The elect rons 
in a region move so as to diminish the 
electric field beyond that region This 
partial shielding effect increases with 
increase in density of *l*ot*onB. ™> 
change in -dE/dx due to tnls effect Is 
usually small.** 

If the index of refraction, n, Is 
net one.'the velocity of light is less than c 
example, n~1.5 and velocity^ light^ is^ J/2 




FIG. II -3 

In water, for 
c . If the In- 




a, particle produ. - 

its experimental discoverer*** 

8. Nature ^f^e_eouation for -dE/dx. Equation II .10 has 

the form 






^T^ne^or^^ Livingston and 

^,e B afeS e in P defali hy Fermi ^ -,%£&* &$&, l.C 

***Cerenkov , Ehvs. Rev. 3E« 378 lUMJ ' 
o. 261. 



[I 



Ch. II 



Ionization of a Gas 



35 



W) 



30ANTITAT1VE CURVES IN MDNI'GOME RY, 
COSMIC RAY PH VSICS } p 35 a 




10 Mc 1 
FIG. II. 4 



lOOMc 2 ENERGY 



The curve BCD gives the l/V dependence. At relativistlc 
energies V changes little and CD Is asymptotic to V = c. At 
relativistic energies, the log term in (V-/l-f5 2 ) changes, and 
increases as V— >e, giving the rise in the curve from to E. 
At very 1 ov,' energies (region AB) equation 11.10 "breaks down 
because the particle has "velocity comparable to that of the 
orbital electrons in the absorber, and the efficiency of energy 
exchange Is much lower. The particle itself captures electrons 
and spends part of Its time with reduced charge. 

9. Ionization of a Kas . If ionization is produced in a gas 
the ions may be collected by charged electrodes, and the amount 
of charge collected will be proportional to the number of ions 
produced. The change of potential of one of the electrodes will 
depend on the charge collected (and the external circuit) and 
therefore on the number of Ions produced. This voltage pulse may 
be amplified linearly and measured quantitatively, a 3 with an 
oscillograph. A gas chamber for this purpose Is called an Ion- 
ization chamber - " . 

+ HIGH VOLTAGE 

R ' 




LINEAR 
AMPLIFHES 




ZL 



OSCILLOGRAPH 



FIG-. II. 5 

In the arrangement In Fig. II. 5, electrons are collected at 
the top plate. A 'negative pulse,. of duration determined by R 
and the capacity of the ionization chamber and associated circuit, 
is produced at the grid of the linear amplifier. 

It turns out that there Is a close proportionality between 
number of Ions produced and total energy lost by the incident 
particle. For most gases one Ion pair (electron plus ionized 
atom) is produced for each 32- 3^ e - v - l03t b y the particle, (see 
table or following page) . Although empirically the result Is a 

* Reference:; on ionization chambers are: Korff , Electron and 
Nuclear Counters (Van Nostrand), Rossi and staub , Ionizat ion 
Chambers and Counters (McGraw-Hill) . 



34 



Scattering 



Ch. II 



simple proportionality between 
number of ions and energy spent , 
the explanation is very compli- 
cated. Theoretical prediction 
of the average energy per ion 
pair involves: (l) calculating 
the % of all primary collisions 
that lead to removal of an elec- 
tron in order to know how much 
energy is "wasted" on nor.- ionizing 
excitation of the atom; (2) cal- 
culating what fraction of energy 
carried away by primary ionized electrons is used 
secondary ionization. This problem has not been 
investigated , 



En e rgy 


for one ion pair* 


Gas 


Energy spent for 




one ion palr,e.v 


H 


33.0 


He 


27.8 


N 


35.0 





32.3 


Kc 


27. 4 


A 


25,4 


Kr 


22.8 


Xe 


20.S 



in producing 
completely 



Problem: Design an ionization chamber and snecify the char- 
acteristics of an associated linear amplifier so that the 
system is suitable for measuring the a energy difference 
■"■' (UI) and U 2 34 fijll) a d< 



between the U 



decays 



92 1 



a + 



90 



Th 



234. 



90 



qiV, 



-„234 



-2S~+ 



92 



234, 



92 



-^34 



tr- 



.a +9Q Th 



230 



The following are among the necessary considerations: £1) loss 
of energy of particles while still in the emitting substance, 
(2) gas to be used, (3) dimensions and electrostatic capacity 
of chamber, (4) gain and frequency response necessarv In the 
amplifier, (5) rate of emission of partiales by the emitter. 



10. High energy g parti cles lose energy mainly by radiation. 
This effect is taken up later, in II, section C, 3. 

B. SCATTERING DUE TO A COULOMB FIELD 

Scattering due to interation of charged oarticles with the 
Coulomb field of nuclei is distinguished from* scattering In 
which the incident particle enters a nucleus. Only the former 
is treated in this section. 



Scattering due to c 
.all charged particles In 
cloud chamber may have a 
angle scattering event, 
quently, and their track 

/3 — 

1, Classical calcu] 



ollisions with nuclei is observed for 
varying degree. An alpha track in a 
single kink, indicating one large 
Electrons are scattered much more fre- 

s are as shown: 



o< 



~N 



screening of the nuclear 
the force on an incident 
Coulomb force ^%&yOl % 
and J- the charge of the 
heavy compared to the oa 
system is almost at the 
and , the angle of defl 



ation for single scattering. If the 
charge by nearby electrons is neglected, 
particle due to one nucleus is the 
, where g is the charge of the nucleus 
particle. Assume that the nucleus is 
rticle so that the center of mass of the 
nucleus. Let b, the impact parameter, 
ection be defined in Fig. II. 6, p. 35. 



For inverse square forces between particles, classical 



^Rutherford, Chadwick and Ellis, Radiations from R adioactive 
Substances , p . 8 1 . ' 




Ch. II 

CHARGE ft' 



Scattering 



35 



NUCI-EUS OF 
CHARGE 2 



MASS M 

VELCC 1TY V 



FIG. II. 6 
mechanics gives the following formula* 




TaN 2=^W* 



11.17 



This formula is valid at non-relativistic velocities, V<< c. 
A relativist ioally correct version of 11.19 for small angles 9 
is given in the paragraph containing 11.20. 

Exact quantum mechanical calculation gives the same formula 
provided the nuclear field is exactly a Coulomb field. Both 
classically and quantum mechanically the formula is valid only 
If the distance of nearest approach of the particle to the 
nucleus Is larger than the nuclear radius . 












The cross- section for scattering of the 
at an angle 9 in the range 19 is defined to 
_L to the Initial path of the particle such 
passes through this area It Is deflected tay 
Since for given particle and nucleus, b is a 
the area corresponding to a given & lies at a 
and has magnitude dC% = 2tt bee) db. Substltu 
of the corresponding angle 8: do^ = 27tb(0)b 
the element of solid angle 27T3ln9d9 to find 
unit steradian, and substitute for b(e) Its 
11.17 for b. Then the cross- section per uni 
is 



incident particle 
be the total area 
that If the particle 
an angle 9 in d9 . 

function of only-. 

certain radius b{9), 
tins for c In terms 
1 (ej&fi . Divide by 
the. cross section per 
value from solving 
t solid angle at 



dcr 
cLcu 


_. i (e a H 3f 


I 


4\M V 2 / Si* 


&% 



11.18 
Note that most particles are scattered at small angles. 

A relativi stically correct equation for 9 as a function of 
b when 9 is small can be derived easily, using the same argu- 
ments used to derive II. 3. Since now v:e deal with nuclear charge 
2 and incident particle of charge ^-* , we must multipy II. 2 by 
the nuclear charge In order to get the transverse Impulse Impart- 
ed to the incident particle in the collision. This gives 

AJo=^£^ 11.19 

v yb 

If As la the relativlstic momentum of the incident particle, 
angle of deflection Is very nearly Z^Jp/Jp , if &Jp«Jp. 



the 



e 






(e 



sma 



11) 



11.20 



If we put fsMV (non- relativi stlc) this becomes identical 
to 11.17 when 6 Is small and tan 9/2 a^ 9/2. In these formulae 

*See, for example, Lindsay , .Physical Mechanics , p . 76 • 



^ 



36 



Multiple Scattering 



Ch. II 



b is limited to distances from the nucleus within which the nu- 
clear charge can "be felt, i.e., has not been screened ay nearby 
electrons. 

2 . Multiple Scattering. Particles , particularly electrons , 
are deflected many times in passing through a foil of metal. 
The net angle of deflection, denoted @ , is the result of a sta- 
tistical accumulation of single small scattering events. The de- 
tailed theory is complicated . A simplified treatment will be 
given here. We assume that no paths are complete loops, as in b 
FIG. II. 7. It is plausible and can bo shown** that the values of 
© for many traversals are distributed about ©-O according 
to the gausslan lay;, i.e., probability f or © in the 



For small 



*s ©f 



Lty 
scattering angles , 



e range d.® is 
11.21 
® P = Z 0^ 11.22 



Dp is the net angle of deflection for p collisions 
and the bar means the average for many such traversals of 
the absorber. 

Since statistically the individual events do not differ, 

~6? =&* II. 2 




j 



FIG-. II. 7 



(gf=2TrND 




Using II .20_Tor 0, 
averaging over Q 2 for 
values of b from b n j_ n to 
bmaxj an< 3- summing for all 
collisions in the length 
of path U, we obtain ***» 



W )uh 



11.25 



of the foil. 



For thin foils, D differs little from the thickB.es 
N — no. of atovns uer cc. 

Due to screening of the nuclear charge by electrons, the 2 
felt by a particle depends on b, therefore Z in the integral is 
a function of b. We take 2 outside and adjust for the error by 
choice of b^. Assuming the absorber is so thin that _Jp and V 
do not change , 

Choice of limits: b max : The equation 11.26 would be 
strictly correct if at distances beyond b^ax the screening of 
the nucleus were perfect; i.e., no scattering, and for distances 
within b m ax there were no screening at all, i.e., full value of 
2 uere felt. No such boundary exists, since screening in- 
creases gradually with distance, but fortunately the log term 



* E. J. Williams, Proc . Roy . Soc . A169 531 (1939); Rossi and 

Greisen, Rev . Mod . Plrys . 13 249 (1941), 
** Rossi and Greisen, I.e. 
*** This is shown in Appendix II ,1 
«*■»* Shown in greater detail in Appendix 11,2 



Cl1 ' T1 Multiple Scattering 

is not sensitive and we mav cut* 

l_ Bohr radius q 



gS 



V* 



37 



11.27 



me factor ? a takes into account the variation of the function 
radius/f ) e4€ " Z ~ (electron charge within sphere of 

The limit "brain effectively adjusts 
the maximum angle in a single scattering 
process. Since we are not "counting any 
values of © > i ,we may impose the rough 
restriction that £?<1. This glims 



--U&HT ATOMS 

HEAVy ATOMS 




2^e ! 



FIG-. II. 8 



!*>« V-^o 



(from 11.20) 



11.28 



Otner considerations may govern b^ such as (l), the finite 
size of the nucleus: b mln >A / 3 1 . 5 x 10 ~13 j frora ePuatlOT] 1A 
? G L° ;; UJ ' 3ize of _™ye packets of particles in tbe colli- 
f 3 ° n **- Tne result for @* using our choices for b min and. b^ 






Jn 



a Vyb 



11.29 

The result Is not sensitive to the choice of b mln and b max ; the 
log terra is of the order of magnitude of 10. JuTae to thf°|« fac- 
tor, scattering increases rapidly with weight of the elements in 
the absorber, even for equal atomic densities. n 

tivi^^^S^^C f, X f kinetlc energy). In the extreme rela- 
tivists range \jjp*^ kinet i c ene rgy . The re f o re , roughly , 



w 



KINETIC ENERGr 



11.30 



This formula gives the :nost important characteristics of 11.29 
Evidently a proton and an electron of the same energy wil n be 
ff^ ed a °? Ut the . sanie amount. "But the range of the electron 
is ,uch greater, ana therefore, experimentally , large anrle 
scattering of light particles is more prominent than of heavy 
particles. . ' J 



Nuraerical formulae: 



ELECTRONS 
IN LEAD 



&' 



£*io e D 



(KILOe.v.) 2 



11.31 

ladiaT^^f' 311 S'^aS 1 MeV el « ctron ^ delated about one 
radian in passing thru 10^3 em. of lead. 



Electrons 

iw MK (B) 2 



j 70oo D 
'(kilo e.v.)' 



11.32 



This is the radius of an atom of charge? using the Thomas 
Fermi model; cf. Lindsay, Physi cal Stat i st 1 c a ■ ^ . 226 

tK^ ar V ls ?" s ^ in T.dlliaras, I.e., and rn^re briefly, in 
J&nossy, Cosmic Rays , para, 173. J ' 



38 



Passage of Electromagnetic Radiation, through Matter Ch.II 



scattering in the mind, 
than the thickness of t\ 



Lnd absorption must be interpreted with 
The true oath lenght mSv be much longer 
le absorber. The equations for range of 



particles applied to a homogeneous beam of non- scattered parti- 
cles passing through an absorber would yield a curve of number 
of particles emerging vs. thickness like FIG. II. 9a. But ex- 
perimentally, FIG. II. 9 b ^s more nearly correct for electrons, 
and the reason for the difference is straggling due to scatter- 
ing;, f NO. EMERGING 



f 

NO. IN 

ORIGINAL 



-^ RANGE - 




THICKNE5S 



FIG. II. 



There is a problem on multiple scattering in Appendix II. 3 



C. Passag e of Electromagnetic Radiation through Matter 

The] 
;d Oj 

ill 



There are three processes by which photons are either ab- 
sorbed or lose energy: 

Photoelectric absorption 
Compton effect 
(3) Pair formation. 



1. Pho toelectric a bsorption. The electrons bound to an 
atom may be grouped into shells according to their binding 
energies. The Innermost shell, the K shell, contains the two 
Is electrons and is bound with an energy of about Ry (Z-l) 1 -, 
where Ry is the Rydberg, about 13-5 e.v. The next shells. are 
the L and M shells. Electrons in the L and M shells are bound 
With energies roughly Ry (2-5 ) 2 A and Ry (Z-13) 2 /9, respectively. 
These last two numbers are very approximate, since the energy de- 
pends on screening and the shape of the orbit. The last filled 
orbits of the atom are the valence orbits producing optical 
spectra. Beyond this, the states are unfilled. Provided its 
energy is sufficient, a photon may remove an electron from any 
of the shells and leave it either in one of the previously un- 
occupied bound states, or in an ionized state. As the frequency 
p of Incident photons is increased from zero, they first become 
able to excite electrons in optical orbits of a feu e.v, binding 
energy. This loss of e-ergy is the first contribution to absorp- 
tion of photons by photoelectric effect. Asj/ is further increas- 
ed, electrons in deeper lying orbits may absorb energy. When V 
reaches the binding energy' of "a particular shell of electrons, 
there is 'a sharp rise in absorption. A rough plot of the ab- 
sorption as a function of V is given in FIG. 11.10. 



The point 
ab so rot ion 



Lt whl eft th e a h so rp t i on change s abrupt 1 y du e to, 
sav, absorption by K electrons is called the K absorption edge, 
and is at the energy at which K electrons are lifted to energy 
levels > 0. Energy is the energy 
zero velocity. 



of an ionized electron with 



sub 



The L absorpti 
■shells 25, 2Pk 



or. 



edges are three-fold, since the L shell has 
md 2Pa/ having different energies. This 







Oil. II 



Photoelectric Absorption 



39 



energy splitting, is due to (1) different screening of the nucleus 
for S ana P orbits, and (2) electron spin-orbit interaction split- 
ting of Py and Py . 



-Z- 
Q 
fc- 

Q- 
CC 
O 
u~> 
CO 
< 




K EDGE 

DECREASE -AS p> 



^B.ET. f k ORBIT, 
FIG. 11.10 



■ 35 KILO e.v -for TIN Z=5Q 



Tne cross-section for photoelectric absorption by one K 
shell electron (for transitions to the continuum of energy >0) 
has been calculated by quantum mechanics, assuming a hydrog<=n- 
like wave function for the electron (Is) . The result is* 



(3 (jW) ^er tatlectvon = 



I28TT e' 



3 e -4scot-'e 



m& zy+ |— q- 



•zve 



11.33 



where Z^ 



.. is the frequency of the K absorption edge, 1/ is the 
frequency of the photon and is greater than l£ , and £=\Tj^ 

The following simplified formula holds for V near to t^ : 



&fe 



(5 p (tV) jaer Is electron «~ 



S.3:' 'O'^JZJ^ 



z z 



Mtteir to 1A. 



11.34 



The mass absorption coefficient A is defined hy the relation 
(fraction of photons not absorbed when passing through thickness 
§ m g-cm tJ of absorber )se4>k*f). A s i m ii ar equation defines the 
absorption coef f icient x . i.e., the fraction massing through 
thicxness x m cm is equal to ~-xx 
" fc; 



Problem: Derive the relation between >u and cr per atom Cal- 
culate^ for lead. y 

Consider a beam of/n, photons-am -2 - sec -1 impinging on a sec- 
tion of absorber of thickness dx and cross-sectional area S 
An area B given hy B = V ( cm^ ) X N (cm" S ) X S (cm 2 ) X dx (cm) 
may be considered to absorb all photons hitting it. B£s. 
The number of photons absorbed per second is (B/s)^ , The 



emerging team has density^ per cm 2 per sec' 



g <ffa 



= -dHcbx 



sk~sw e 



—trMa. 



A STL 

The fraction net absorbed in a finite distance x (cm 



^ = P^ N ^ = ^^f" 



m 



= C 



is 
by definition. 



Heitler, quantum Theory of Radiation , 2nd ed/p. 124 



■■-■ Photoelectric Absorption cii. II 

But r*»y where /> is the density in g-cm-3. Therefore 



^ = f(^Y)"-^ m W™ ^ N = £* ,o 23 



11.35 



£^ f b * °^culated by equation 11.33, multiplied by the num- 
ber of K electrons P er atom, i.e., £, an d^ found by II 35 
Tills calculation ignores the contribution^! the L^M N etc' 
lit) ?h, f C r?i lnS t0 Rutherford . Chadwick & Elli . ;, 

4 n ^ v S ? e ii v aCC T ° T U ? tS ** -a* of absorption for photons 
ene?Sy 5 19 E 3 ^ ^ S e ^i° n 11.34,^/ for lead at photon 
energy- .19 mc la -.,-6 en--g X, Actually,* is— 8 In 
1 x nr, 1 ng^ , 5 . 76 , we i gno red ab o r ?t i o n in other than K shell. 



tth change "of K 



The sharpness of the absorption edges nohes pee 
methcu of measuring the frequency of ^ rays of low e 
J edges yary for different elements, and one may det 
tween which two K edges the unknown nhoton erern; 11 
lng the sudden change in absorption ~ ' 
of the absorber used. 



sible 

■■ =1 LiJ ■ 

ermine 
e s by- 
edge 



The 

e- 
ohse re- 
location 



a* 


C 


u 


S 


n i 


D l 








\\ 


* 


1 

\ 


1 
\ 


.Oc 


>:5 


.0 


2 


.& 


\ 
7 . 


9 



rNERGY IN UNITS OF "mc* 



LOCATION OF K ABSORPTION EDGES 
PIG. 11.11 



Proolem: Giver, a source of 05 kilovolt H radiation. What 
absorbers would you use to bracket this energy and thus~ serve 
to determine it? Determine the thickness to be used. Qlve 
r™n Q8 l0n t0 thS Pliability of the chosen absorbers. 
Compounds may oe used. The absorption of a compound is the 
sum of the separate absorptions due to the constituent 
element s. 



3 . 
'-&V in. 



Comoto n scattering . Photoelectric ab so rot ion decreases 
r-eases. Soon after the K edge is massed the oredomlnant 
process removing energy fro.-, a beam is scattering by electrons 
Si iV?\ true abso ^Ption, since most of the energy Is ro? 
absorbed but rather sent in a different direction"? In the low 
energy region h V « m c 2 , scattering cross-section aln be cal- 
culated classically and is given by the Thomson foxmSia: 

elec+ror^Vt^i^^^f 12 ^ ^"t^nagMtt* wave impinges on an 
AsluS The JtJ leCt r n / 3 SSt ln m ° ti0n and therefore radiates. 
,^T J*f el ^ctron is free, a good approximation If ^ of the 
is tha ^rT h h^G^ than the frequency of the electron I? A 
is the acceleration of the electron, " 



e£ =rmA 



PLANE POLARIZED 



A^ 



A = 



PHOTOH 



K 



■ar L 



FIG. 11,12 



"" f ELECTION 

ELECTRIC VECTQK £ 
OF £, M. WAVE. 



Ch. II 



Compton Scattering 



*L 



The instantaneous power radiated from a moving electron is* 
dt " 3 c3 11.36 



2 e P T 

3 c 4 /w>* 



In this case the average rower ia j£ 

ar 

This energy taken from the incident wave and re- radiated may- 
be expressed in terms of a cross- section C% per electron defined 
so that 

_, (total energy radiated per 

Oj X (Intensity of Incident beam)= un it time by the electron) 

Q^cra 2 ) X I (erg-cm^-sec"- 1 ') = yy (erg-sec^ 1 ) 11.37 

This definition is equivalent to saying that all the energy in- 
cident upon a surface Cfr i s absorbed and re?- radiated. The 
intensity of the incident beam is 



I --T^sz. (evtf-o^-sec ') 



Therefore 

U T — r _ lM = T' r e' = c - 66 x 10 "• -'(cm ) ( per electron ) II. 38 

where r g Is the classical radius of the electron, 2 .8 x lO" 1 ^ em. 

The Thomson cross- section is evidently independent of fre- 
quency. 11.38 breaks down when h.p is near or above mc =^^Mev. 
Then the phenomenon must be treated as Compton scattering. In 
the Compton effect, an incident photon \-JJ collides with an 
electron. Compared with the energy of Irz* , the binding energy 
of the electron is negligible. The scattered photon suffers a 
change of energy which can be calculated by imposing conservation 
of relativistic energy and relativistic momentum on the collision**. 
The result is 



■"Scattered — A incident = X^T \\ -Cos ©J 



11-39 



nV Incident 



Vixj sc&ti&red 



electron 

The frequency of the scattered photon is less than that of the 
primary* The factor h/mc«^ c is the Cometon wavelength, 
0.024 x ID -8 cm. 



The cross-section for Compton scattering Is given by the 
Klein-Nishina formula***: 



11.40 



^Abraham- Becker, Tbeorie der Electrizitfit , Vol. II. p. 73 
**Thi's is done, for example, in Rlehtmeyer and Kennard, Intro - 
duction to Modern Physics, 3rd ed . p.. 533. 
*'«*Derived in Heitler, I.e., p. 149 




42 



■ 



Compton Scattering 



Ch. II 



Problem: F ind asymptotic Bxnreualora for II 4n nZ i ^ 

enemies and for higJl energle5 _ ^ ^11 .40 'rouS^! " 

The asymptotic expressions are: 

.Low .energies: C = a T [ 1 -Jtw- + ^**+ ... j , <*«! H.4I 

HiSh energies: eg « fc^ka*-^) rt>>1 



The Klein-Nishina formula is plotted below;* 



\AZ 




Direction of 

POLARIZATION 



rorJa n ^I?3 r 8 d L1Sf1L 0n a„4^ r / d T st en ?^i r where the Thomson 
13 that due to a radiating c^atlica rtrll «* 1 « a ° f *»**tad ener^ 

ft'VS ^ ^ the an S le between 

iBBrit S 4 ■ - Primary radiation 

azW* P f 1 ^' an ai ™3* over the 

5ffi£*}* ^* tne maxima of scattered 

CP J^p " hen ^f approaches and ex- 
tlAtA radiation is predominantly 




'INCIDENT 
RADIATION 



FIG-. 



¥+9^30° 

11.13 



lm,enSt e y S ifS foTow'nJ ■" ""S^ ai ***»M*» of scattered 






Jl a | +^(1-^.9) J 



H.43 



Sw/J Tn?I * thS 1 J?S 1 ? f* Ate the direction 9^ ■ a 
cident radlatlonT^, Sein \i2^f ^ <* P laM polarized' 1?- 

from integrating this eqiaSon! ^ II,4 ° WOuId res ^ 

* From Janossy, p 05mlc Rays ( (0xford)i 



Ch. II 



Radiation by Fast Electrons 



43 



3 ■ Radiation loss by fast el ec trons {ore m s st rah lung ) . 
This paragraph should coin© under II " XJ sub-section 10, butls 
placed here be cause our calculation makes use of the Thomson 
formula for OV , derived in the last sub- section. 

Consider an electron of velocity V ^=c -passing a nucleus 2e, 
We calculate the energy given off by the electron in the forn of 
photons when it is accelerated by the field of the nucleus. 



FAST ELECTRON 




In the rest system of the electron, the nucleus Ze moves with 
velocity almost c. The electric field £ of the nucleus is con- 
tracted, and the associated magnetic field H is X. to £ and of 
about the same magnitude. Therefore the moving nucleus looks to 
the electron like a plane electromagnetic wave. This will be 
discussed more fully shortly. The wave of photons representing 
the nucleus suffers Compton scattering by the electron. These 
scattered photons, when viewed from the rest system of the nucleus 
(the laboratory system) appear as the photons emitted by the in- 
cident electron. According to this brief summary, the calcula- 
tion has the following parts: 

(a) Lorentz transform to the rest system of the electron. 

(b) Calculate- the characteristics of the Dulse of electro- 
magnetic radiation that represents the moving nucleus. 

(c) Find the density of photons corresconding to the pulse 
of electromagnetic radiation. 

(d) Calculate the probability of scattering of these photons 
by the electron. 

(e) Transform back to the laboratory frame. 

Throughout this calculation we leave out numerical factors. 
Their inclusion would obscure the Ideas involved. The correct 
numerical factor is Inserted at the end. 

(a) Let p = %«1 I i* IA/I-/3*' is large 
the electron in the laboratory frame is Ymc^. 



The energy of 



LAB, FRAME' 



ELECTRON 
FRAME 



Je 






FIG. II.15 




44 



Radiation "by Fast Electrons 



Ch. II 



In the lac frame, denoted by °, the electric field of the 
nucleus is \£°| = %*/&. In the electron frame, denoted *, Its 
perpendicular component £* is Increased by the factor H\ 

The magnetic f ield °&* seen by the electron in its own rest 
system i s _L to both £* and the velocity vector. Denote this 
field by ^ = | -A*\ . Then* Kj = € L %0 *^S ^ *| . V , 
Therefore y* = e * ( as ln a plaa@ electromagnetic wave. 

(b) The reel on in which the electromagnetic field is not 
zero has a length, In the electron frame, of about b/v , the 
l actor g coming from the Lorentz contraction 

SHAPF OFPUL5E: SPECTRAL DISTRIBUTION; 




t-0 when the electron is in wave- front Diane representing 
■nucleus " ° 

FIG. 11.16 

The spectral analysis of this pulse raay he done by assuming 
for simplicity that it is a Gauss! sn of time width b/fc,; 

e, -x, cc e 2be Z1 . 44 

SSto^iSSli^? 11011 ° f C °^° nent ****»*« l- -then 

di ^Lsrtr.WiL^^ gst^Tr p 2 e ^^sr ic 

volume is dven bir S +X ari p- 3si I ?cY r _, 

n rrn ™p f^f x Y? n D/ Tjr;" p" V*F*7 ' , . Tae volume per unit 

area of the disc ecuals the thickness = b/y (cm Therefore the 

energy oer unit area Is ° ' re 1 ' ne 

(c) Approximate to the gaussian spectral distribution by a 
rectangle of width Xc/b ***■• This means that the energy carried 
by photons In the frequency range Ay* is simolv prooortkmal to 



*5h raham-Eecker, Vol. II, p. 48. " ~~ ~ 

**Strattoh, Electromagnetic Theory , p. 290. 

»**it will turn out later, (just before 11.54) that only frequen- 
cies in the electron system less than p* = mc^/h will be inte- 
grated over, and that tyj* » S**g .<< '|^fc , Therefore the portion 
of the rectangle used will be at a relatively flat oart of the 
par s si an , a s sho wn i n F I g . 1 1 . 1 7 . 



Ch - II Radiation by Fast Electrons 



<-o 



V. 



MSI. 



^C 




therefore the energy carnled by 
photons d: this rr n - :e a*,* ^ 

m a at distance b, i§ P 

Az--*b ? 2 e^V 

The number of photons tier ~t 2 in 
the range A^ is equal to this 
XCenergj per photon^ 1 which is^)" 1 
Therefore the number of oho tons 

£X + m l;V h ? ^anoncy" range 

T.r_oe bis giver, bv ; 



A^*at the dial 



FIG- 11.17 



S&hS^'b 



b 3 ^c hv" 



o V y* > & 

b 



fj. i ™ f W tor ^ > * — 

I aj The mean number of o,„ t , Q „, ^ D 

frequency ^* occurri fi|i? the S SKTl" *** ?hotons of 



GQ X (density (per cnT 2 ) o 
Therefore 

No. of scattering everts 



photons of free. V* ) 



for rre a< ft~^ ^Wi^L *V A£* , « 

01 h, am for one nucleus / ^h b a ^* c ^ J 



II .45 



offfl^!S cross-section per nucleus be £fe*)*ft/*- ■ * 

whi^ Beyond b max ^X/^* ths f T&QV[miir y ,* (Y 

which we are commit 1 -ir- vr TJ *\ \ * ■. ^-^^.„/ ^ (for 

is the case and use max > W' - e assume this 



u min 



'max 
is subject to seve: 



limitations. 



II .47 



Sends on the d-.-m-i n,,-i~~-.,~ "~ """"'" Ul " L,J - u:!B ' Which is dominant de- 
must bel Lt " ^ The ni0st ^Portant is that b , 
ing the ^«1^S^;^^ n ^5«±«»« ^ Packet' reor^e^t- 
the Compton wavelength i * wl le ? S , h for the electron is 
velocity c/^ ,ithre S ^t to +X ^ *** *** electron have 
4e Broglle vlvelength £ *° d *??,?** S ^ m > 1*1 a giving it 



velocities. Taking K^_ - f 
>-- 1 "mm - A 



Prna », 
tarn,-. 



a 



',et: 



4 = 137H"' 



11,43 



tm s pomt is w«asxs sjfgs/itif^^: 1 ^^^^-. 



m 

More nreciso 



Radiation by Fast Electrons 
acuiation leads to 133 Z' for II. 45. 



^v*)av*^^*<1 c JU 



173 
§1? 



5,0 



SVSt« 

Fro: 



( e ) T r a n s f o m. ":ac]i to the lab 
the photon changes from ~p* to ~i/ a 



Therefore 



Introducing this result into 11.49, 






'-Kh=-nr^e 



Ch. II 
Therefore 
II. ho 




The frequency of 
special relativity* 

11.50 



11.51 



et; 



11.52 



The factor 4 /%. results from exact derivation. The cross- section 
per nucleus for scattering at frequency J/ in the lab system is 



^•JA^Z^/A-H 



11.53 



where e~/c)i, the fine- structure constant, has been renlaced by 
1/137 ■ 

Before integrating to find the total energy loss by the 
e 1 e c t ron , we e xam Ine the limits of validity of thi s re suit , In 
11.49 the maximum value for Oc was used. This breaks down at 

[IV* = mc 2 , This defines the limiting value of I/* , namely Z^" . 
Corresponding to 7-i* are two values of ~U° , one for forward and 
one for backward scattering in the electron frame. (Thomson 
sc at t e r I ng 1 s symme t r i c al , f on;?, rd and ba c k . } The 1 a rge r value 
of ~P a results for a numerator in 11.50 of about two , i.e., 



Ke, = H 



* l+/3cr>vB* 



m 



Therefore the use of the Thomson scattering formula 11.38 causes 
the theory to break down wher- h^°/V = mc'-, or 



But thi s sav s 1/ 



:i!;.X 



enenrv of ele ctron 



11.54 



, which clearly must be 



so , because the electron cannot radiate a photon of energy great- 
er than its Initial energy. 

The expression 11.53 diverges at low frequencies. But the 
enerrv carried by low frequency photons is finite. The total 



TJ' 



per oath lenr.tb dx is 



energy loss to photon^ of freriuenc;. 
equal to : 

(energy ^>er |oinoton) X (no. of nucleytm^X (crass - section "£{V)AV") X &X 

Therefore -(cIeJrao. , the total average energy loss per path length 
dx for all frequencies Is given by: 



ft 4 ". JzjVsO 



11.55 



* cf. Abraham-Eecker, Vol. II, u, 312. 



Ch. II 



Pair Format i 



4.7 



^-^^f^ 



II 



'.■.'".lore 






! 



nzC = energy of s ].e< 



tron, 



ete^fT^t/lf^ 71 ^ '~ Xve an a FP re ^bl e amount of its 
rro^^e^oraSo^ffif^'l^? ^ 1 ***** l0M ** *** Wid ^ 

(fiE7dS) rad is evidently proportional to energy. It Is the re - 
hi 3 encr^ief *" C ° SmlC ra7 P h 7 3ics w ^e particles have very 



We may v/rite 11,56 in tenai 



Define 
Then 



X- 



-m-^^Y* 



(3e) = _ e ^c 

x -vat, F- s 



RID 



or 



°- ^ , the ra diation length , 

11.57 



jar 



For 3f«.4 , I/2.7- 

c 



-£ 



Roughly , 



rn 



E^E^e -" 11.58 

the energy Is dissipated, on tho average 
whs re C is nearly constant. 11.59 



Jk «C 






Radiation loss r>er g-civT 2 traversed 
is greater for heavy elements. The 
ratio of radiative energy loss to 
Ionization loss, for electrons, is 
given rouahlv by* 



^Ens 



Material 1 n lEnergy for 

. ccsjiton ~ djxjiot-i. 



otX / RAD. 



, Air j NTP 
J Alurni:iu:i 
Lead 



"330 rn: 

9,7 cm, 

.517 cm. 



ISO Mev 
5S 
7.0 



? * ( Me vj 
800 



H.6O 



(Thickness of the atmos- 
phere is equivalent to 8 
fen. of WTP air, or about 
SOJJt ) 



+ jor , Js e r^ rotation . We return to the discussion of absorp- 
tion 01 electromagnetic radiation in matter. The mos+ limoS 

S^i^S Ve T 7 r- lGh ener ^ ^ectroma^tir^atWirby 
SSnKSi ?" tMS PrCGBSS a ph ° ton diaappeara and a ros- 
Dir°c's ^;\ n 4ifi r ° n appear * This is t0 be underatood only by 
is slrr^ii^L ? 1C ™y e /«^hanics. The following discussion 
is aimplliied and qualitative**. 

According to the relatlyJ^Mctheory of the election an 
electron tea energy + xfu^WTZX^ mJ. \. Le ^~ -' ^ n 

lj -yinifi ) + p c . This equation permits 

"This appears .inverted, in Lecture XIII of Los Alamos Reoort #24 
d\lt ?£ prin ~ Gd version. Also inverted in page 72 of fia'setti. 
Trie theory is given, for examole, in Heitler, I c 



48 



Pair Formation 



Oh. II 



I 



negative energy values. The energy spectrum of a free electron 
looks like Fig. 11,16. 

In Dirac' s theory, practically all 
negative states are filled at all points 
in space. A vacuum is then a sea of 
electrons in negative energy states. The 

presence of this charge is rot observed +<mc : 

because it is uniformly distributed. 



A photon of sufficiently high energy 



■- 



I 
i 

FIG. II 



18 



may lift an electron from a negative 
energy state to a positive energy state 
The energy threshold for the photon is 
2mc 2 , since for a free electron there 
are no states between -mc 2 and +nic 2 . 
Physically, this means that the photon 
must supply enough energy to create two 
particles of mass m. Momentum must be 
conserved and this requires either that 
the negative energy electron be near a 
nucleus or an electron, i.e., not free, or that two ohotons 
coming from different directions coalesce and lift an electron 
from a negative energy state*. If the electron is near a nucleus 
it may occupy discrete states just below +mc 2 . These are within 
a few e.v. of 510,000 e.v. Strictly, then, the threshold for 
pair formation near a nucleus is 2mc 2 - (binding energy of 
electron). This is of no importance because binding energy «mc 2 
and because transitions from negative energy states to the dis- 
crete part of the spectrum are improbable and not yet observed. 

The result for cross- section for uair oroduction near a 
nucleus Is calculated in Heitler** and is 



II.61 




In the extreme relativistic rai 



en 



rmr is independent of energy, 



For other energies the situation is more complicated, and is 
described in Heitler. 

(A simplified procedure similar to that used to comnute the 
radiation loss by fast electrons may be used to comoute the cross- 
section for pair formation at very high energies, if the orocess 
of collision of photons in a vacuum {h v + hv-*e + + e~J is known 
This is outlined. Consider a uhoton of \iv ^ 1000 mc 2 or so. 
Transform to a new coordinate system in which the nucleus moves 
very fast, almost, but not quite, c. The ohoton seen in the new 
system has reduced frequency. The nucleus looks like a wave of 
photons '(just as in subsection 3). Oive the nucleus such a vel- 
ocity that its photons have the same (reduced) frequency as the 
incident photon seen in the new coordinate system. The" process 
i ■:-■, the simpler one of collision of ohoton s in a vacuum, ' 



h v 



W- 



+ e" 



) 



shown in Appendix II. 4 that momentum cannot be conserved 

_r f OTVPra.t, 4 r nn Vnr r->no 1 onl a+ aA v-o, ,-,+ ,-. .« 



Xt is onunu j_n .tiuufciiJCLx j.1. ^f t,nat. momenta 
in pair formation by one isolated ohoton, 
*#Heitl g;r, i.e., p. 200 



Ch.. II Pair Formation 

We may define a mean free path for pair production 



49 



Jp^ 



I 



II .61 



Then the decrease in intensityynof a beam of photons will follow 
the equations: 

7C 



dm b- „ £* /jt =^ Q e ^ 



11.63 



/n 



^ 



4- 



i 



23 -Z a N _2 S 183 
5 T37 yCe ^-^ 



11.64 



This is analogous to the equations for radiation loss by an 
electron, 11.58 and 11.57 

e -i (H.58) 



E 



J* 



& 



Evidently 



4m z H„a j U3. 



-*-•£-< 



(11.57) 



11.65 



COSMIC KAY 



5. The phenomena of radiation and pair formation are 
responsible for the shower phenomenon in cosmic rays*. The 
shower is initiated by an electron 
haying energy of the order of 
billions of c.v. By radiation it 
produces photons of comparable 
energy. These photons produce 
electron-positron pairs, each 
particle of which radiates. The 
energy is soon divided among many 
particles. When the energy of a 
particle reaches the point where 
ionization loss predominates, the. 
shower stons. 



Me sons have ma s s about 
The factor (e 2 /mc2)2 appears 
formula for radiation loss, ^ 
meson, this factor is(l/2.Q0r 



00 m. 
in the 
For a 
as 




large as for electrons, hence energy 
loss by. the meson by this process is 
much smaller. A meson looses energy mainly 
a meson of some billions of e.v. ionization 
the meson can penetrate several meters of 1 



FIG. 11.19 

by ionization. 
loss is small, 
ead . 



For 
and 



b. Summary . The phenomena of absorption of photons can be 
summarized in the following graphs. 



* The theory of showers is given in an article by Rossi and 
Greisen, Rev . Mod . Phys . 13_, 249 (1941 ) 



' D ° Absorption of Photons 

GENERAL FEATURES OF ABSORPTION AS A FUNCTION OF ENERGY: 



Ch. II 



PHOTOELECTRIC 
ABSOF^PTIOW 




PAIR FORMATION _ 



.01 




COMPTON EFFECT 
f (KLEIN -N1SMINA) 



' l0 ENERGY^nric' 1 UHTO 100 

FIG. II. 20 
ABSORPTION COEFFICIENT FQ* PHOTONS iN LEAD, AS A FUNCTION OF ENE^T: * 



h 
z 

Ur 



U- 
U 
O 

o 

z 
5 

Q. 

B 

8 
8 



\ 




Kw - limits nrrLC 1 



FlS.II. 2.1 



MATERIAL 



Alur 



Colpf3ey 



Lead 



Location of tninimum in 
total obsor^tion 



-35 MEV 



~IO 



3M* 



Thickness F j in 5/01' 1 f-w l f&f*$Q 
dedu ct mn an tntemi-iy 



■ 47 &£m 3 at -35MW 



•31 



-10 Mei 



-15 



-3J£m< 



Problem: Suppose e ^gos-tUg ray electron of energy ld00~lFc 2 ~ 

tut the electron lec.ves the plate ^conroanled by a nair whose 
total energy (electron plus positron) is larger than^OO m^ 7 
assume tne pr.rtic_Les chan-e direction nenlifri t>ly . 

Use 11.53, wri+lnc It in terms Q f^ ■ grf'^ ! 

To ? ;et the cross- section for all the nuclei ^thickness dx, 
by U dx, -ettlnr 



multiol: 






feU ^ 



* From Reltlt 



216, 



Ch. II 



Aftiendix H.l 






cro 



53 S9 ; u ^Sj hotons ir the ^ 5o ° - 2 *> 1000 BQ * 



J WjT - 






A ^°_ .693 j 
~^77 - — ^— OA" 




Will 



This is the nrobabillti tha j - t. wu j 

be produced In LJ tSkSX dx^ " ° f ener * r > * 

The probability that a photon ,^ 

duced at X will then form a ,air 

fron 11.63. The pair Produced 
Will nave all the ener 3 of the 

that both- events will occur Is 

[using H.65 For lead in this energy region 

[^^ Ihereiore the probability is O.OO75 

the scattering anrl e fn-n - =■ i " "■■ " > Where &. j S 

over .any traveraSs 0? the ^ftrbe^Tt^ ^ ^ lc ***«> 
alated angle between the IniUal df^.ff^ ^ b9 tKe net a ^uin- 
particle's velocity vector \^ ?£ Q ]J?^™* ^d the 

the traversal in ah&<?ti <-m a lii + re , " L1 . e -^.+ i collision o^ 
next paR e. W * atwm ' A Sorter derivation appears on the 



NOT IN PLANE OF 
PAPER IF i. ^o 




TRIANGLE 5 

4+1 



JM COLLISION-) 




A r -'V= ,to -^ + - l 



I^TIAL DirecT.ONTOWH.CH © k IS MEASURED 



FIC-. 11.22 
This is a recursion^^I 2 ;, L d :^—^ *** respect to f, 



(Cos© p )=rTTc 05 « 



II. 67 



ii™ S^Jiri, 8 * P^ 1 ' '"'- "^- «- -"•■ 



'" #•- #(.-£?« 



ST 



£ ? S j~ P**4 gj»- a- jf |s 






52 



Appendix II .3 



Ch. II 



Shorter derivaxion of 11.22: ,„„+„,.,, 

We make use of the fact that small, angles are vector a. 
p 

Then Op = ?^. -i 



S£f 1=x 



$ =£*? *£*•** 



In averaging over many 
■t riu s 



&*£% 



many traversals, Oj is positive as 
e_ — r 
1~ 



often aa it is negative, 






Aonendix II .2 . Deri 

i greater detail. ?, the avsra C e number of collisions in one 
traversal is given fc y 



If f is a function o7 X M& N# is the normalized probability 
den sit;; forX , then the sean of f is 

J"=» fj%j ^fcrj d>K II . 59 

anpl ice. 1 c a I cul at e ©\ S 1 = J [©0=) ] * ttfo) d b . 

The probability density for b, #J Is proportional %a prow.- 
sectional area at the distance b, 2Trb db , to the nuclear 
density N," and to the path length D. Therefore the normalized 
Wfe) I a Z ir lo D N 2 tt b DM 






Therefore 



Finally 



— i, f[&Cb)] 2 2-rrbDN4b 



II. '0 



II. 71 



©*= ZTT 






(II 



Anoendix II.? . Equation 11.26 can be derived by consi dering the 
^ ro lections "of the angles A on some chosen plane and deriving a 
value forV , the mean square of the resultant P™ -ted angle 
Then the unpro jected, true scattering an G ie can be found from MM 

result that ± @* = -q> a (^..<<,1) II * 

The factor £ comes fror. avera G in G cos 2 , the an G le bein G random, 
A "similar relation holds between the actual linear deviation ol 



Ch. II 



Appendix II ,3 



53 



Placement on some chosen pSmT'i!S! ^^ct-ion of that dis- 



/. emergence 

— f^— — I 

SlWpsy of entrance ^fc 



-g7^ 2 = -x 2 



11.73 



(plane of paper normal to Incident 
direction. Lookin 5 toward source 
01 particles) 



In a cloud chamber Efe-oissrrartb 1+ i- a +1 

scat terms that 'la shownf " ls the ? ro J G ^ed angle of 



-^^=^^ used bj danossy 



Problem.. Electrons of ensr-py 

10° e.v. (from cosmic rays) 

pass through a lead nlate 1 G « 

thick Calculate the mean square , 

01 the linear deviation of the electron 

emergent points from the peo- "" 

metrical shadow point on the- last 

the patii is almost straight. 

|^%«^^iS^^^S?S• ,, f ^-the^sur- 
The projected angle of deflexion at ^ n aZ ^ h tn Sle i S £ 
Projection of linear dissent £ 1" "f**^ - *» ^ 




DEVIATION » 



last 
irom the/^sur- 



for colli - 
sion =£; : 



PROJECTION \^" 



OP PAT h v ,. , 

*■ -D 






1= 



& * FNCE TOWARD SOURCE OP 

Particles. 



**^pHj%%^ S 1§^&, bat C^c^ = X5 



^^ff / 3,Mce f* a "d i a r e independent 

" ^ "^ 6 2 P P= averse ho. of c.o|[ hS ions 



Th^un.^ected linear (dlspW^ht^ ^ , will be ^0. 
^ 2 =^D*©* (^0.02 Cffl. USln£ H.31) 



54 



Arrendix II .4 



3h. II 



g4& 



11.74 



A-Q^endix II .4 . Proof that moimnixw cannot be- conserved in hrz 
tion of an electron- positron pair by a sir.yle Isolated photon. 
By conservati on of energy : L 

By oonseri r ation of momentum: 

(The equality holds if 3 + and 3~ f/b ° r1 ' ln the same direction) 

Fro, n.75 (M*^ #c**4L*/e» + MM?^ 

Fro, II .74 M" = *fcV * * ^Vzt^lM^ Pf^^ x 2 

The radical terra is larper than 2yjo +/ f5./o 2 , so both carrot be 
satisfied . 

References: Books and articles referred to ;:ore than once in 
this chapter are listed. 

Ab raham- Be c k e r , The orie der Slectrlzitat , Vol, II ( T eub re r , 

Ed.uards) 

Heitlcr, Quantum Theory of Radiation , 2nd ed . (Oxford) 

Janossy, Cosmic Ray s (Oxford) 

Livingston and Be the , Rev . Mod . Phy s . , £, 245 (1937) (experimental 

.nuclear phy s i c s ) 

Rutherford, Chad wick and Ellis, Rad i at i c n s f ro n Radi oa c t iv e 

Substances , ( C a mb rid ge~) 

Rossi and Greisen, Rev . Mod . Phys . 13, 249 (1941)